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Properties

Label 16.4.68259509250...6097.7
Degree $16$
Signature $[4, 6]$
Discriminant $47^{6}\cdot 97^{15}$
Root discriminant $308.77$
Ramified primes $47, 97$
Class number $40$ (GRH)
Class group $[2, 2, 10]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2220423519641, 553864288512, 541452941068, 52344317495, 25241662310, -1323459410, -831919255, 80128862, -92090702, 15481249, -4073673, 379060, -37712, 1269, 14, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 14*x^14 + 1269*x^13 - 37712*x^12 + 379060*x^11 - 4073673*x^10 + 15481249*x^9 - 92090702*x^8 + 80128862*x^7 - 831919255*x^6 - 1323459410*x^5 + 25241662310*x^4 + 52344317495*x^3 + 541452941068*x^2 + 553864288512*x + 2220423519641)
 
gp: K = bnfinit(x^16 - 2*x^15 + 14*x^14 + 1269*x^13 - 37712*x^12 + 379060*x^11 - 4073673*x^10 + 15481249*x^9 - 92090702*x^8 + 80128862*x^7 - 831919255*x^6 - 1323459410*x^5 + 25241662310*x^4 + 52344317495*x^3 + 541452941068*x^2 + 553864288512*x + 2220423519641, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 14 x^{14} + 1269 x^{13} - 37712 x^{12} + 379060 x^{11} - 4073673 x^{10} + 15481249 x^{9} - 92090702 x^{8} + 80128862 x^{7} - 831919255 x^{6} - 1323459410 x^{5} + 25241662310 x^{4} + 52344317495 x^{3} + 541452941068 x^{2} + 553864288512 x + 2220423519641 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6825950925050758427882178827125973556097=47^{6}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $308.77$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $47, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11773} a^{14} - \frac{4906}{11773} a^{13} + \frac{5623}{11773} a^{12} - \frac{871}{11773} a^{11} - \frac{830}{11773} a^{10} - \frac{769}{11773} a^{9} + \frac{2522}{11773} a^{8} + \frac{3112}{11773} a^{7} - \frac{4812}{11773} a^{6} - \frac{3302}{11773} a^{5} - \frac{2524}{11773} a^{4} + \frac{4720}{11773} a^{3} - \frac{2614}{11773} a^{2} - \frac{1919}{11773} a - \frac{2993}{11773}$, $\frac{1}{3142761642962386217147110608944456691296865126424223371666385105931063911744800277548991} a^{15} + \frac{101096463925096799143590458681625248072069562638166387298412381143112549180419495349}{3142761642962386217147110608944456691296865126424223371666385105931063911744800277548991} a^{14} - \frac{639049178467794877747351193796051694569847353244113023380957925679822413465420612721106}{3142761642962386217147110608944456691296865126424223371666385105931063911744800277548991} a^{13} - \frac{1153498339788283371823882062199195496980632141229709744033547590956425618226419905121318}{3142761642962386217147110608944456691296865126424223371666385105931063911744800277548991} a^{12} - \frac{1190556528900180548836443649659570999101072881852847609196317379768987582636899079417268}{3142761642962386217147110608944456691296865126424223371666385105931063911744800277548991} a^{11} + \frac{77731259707472727085399225818252606834394174851455586672717530260946043607386667015981}{3142761642962386217147110608944456691296865126424223371666385105931063911744800277548991} a^{10} - \frac{1477352007851356148682834571575233439461425271354790170340560722412768803092992394509219}{3142761642962386217147110608944456691296865126424223371666385105931063911744800277548991} a^{9} - \frac{1345534281025662644842401899903720032911667869144412773964188229062944280745482926860640}{3142761642962386217147110608944456691296865126424223371666385105931063911744800277548991} a^{8} - \frac{18761892821583869341534324473634736857833585630523435931317322866485679514609586719135}{51520682671514528149952632933515683463883034859413497896170247638214162487619676681131} a^{7} - \frac{127417029315516675940468367996480539910743788634143878877992869401134388051091275176005}{3142761642962386217147110608944456691296865126424223371666385105931063911744800277548991} a^{6} + \frac{1162760511925498198819137770708660743545257657373649462763090642405424857812744875424345}{3142761642962386217147110608944456691296865126424223371666385105931063911744800277548991} a^{5} - \frac{937213272661227184351892173264941185839591579938462647775800306215178605946363535505834}{3142761642962386217147110608944456691296865126424223371666385105931063911744800277548991} a^{4} + \frac{327174335857680356505654156242261778125962692433133991843812872202944204241244316084866}{3142761642962386217147110608944456691296865126424223371666385105931063911744800277548991} a^{3} - \frac{717290081985086528137234386568923494297211581627128667330589113079389816736873721260818}{3142761642962386217147110608944456691296865126424223371666385105931063911744800277548991} a^{2} + \frac{455922730408802926623867636279611667616149767746087815410885723915344797064391309350859}{3142761642962386217147110608944456691296865126424223371666385105931063911744800277548991} a - \frac{1283015047483088307527242988025362811720331081180291582246368745081171351388217234276072}{3142761642962386217147110608944456691296865126424223371666385105931063911744800277548991}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{10}$, which has order $40$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 888041715537 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.80798284478113.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 sibling: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$47$47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.0.1$x^{4} - x + 39$$1$$4$$0$$C_4$$[\ ]^{4}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97Data not computed